The Special Characters Token for a Quarter-em, or Flexible, Space is &:q.
To display the non-breaking variant of this character, use the token &!q.
Quarter-em, or flexible, spaces behave in exactly the same manner as en-spaces and have the same usefulness. The difference is their width. While en-spaces are defined as always being 1/2 of an em-space, the quarter-em space has a default width of 50% of an en-space (ie. 1/4 of an em-space), but that percentage can be modified in the QuarkXPress document preferences dialog (hence the flexible nickname).
The quarter-em space behaves as a normal (not a space) character would : it will break as a regular space would, but it will not expand due to word alignment such as justification, and it will expand and contract according to the kerning, tracking and horizontal scale settings that are applied to it.